Reproducing Kernel Space Method for the Solution of Linear Fredholm Integro-Differential Equations and Analysis of Stability

Abstract

We present a numerical method to solve the linear Fredholm integro-differential equation in reproducing kernel space. A simple algorithm is given to obtain the approximate solutions of the equation. Through the comparison of approximate and true solution, we can find that the method can effectively solve the linear Fredholm integro-differential equation. At the same time the numerical solution of the equation is stable.

1. Introduction

In this paper, we consider the following first-order Fredholm type integro-differential equation:𝑢^′(𝑡)=𝑞(𝑡)𝑢(𝑡)+¹₀ℎ(𝑡,𝑠)𝑢(𝑠)𝑑𝑠+𝑓(𝑡),𝑡∈[0,1], (1.1) with the initial condition𝑢(0)=0. (1.2)

The equation is discussed by Yusufoğlu in [1], where 𝑞(𝑡), ℎ(𝑡,𝑠), and 𝑓(𝑡) are sufficiently regular given functions. Integro-differential system is an important tool in solving real-world problems. A wide variety of natural phenomena are modelled by Fredholm type integro-differential equations. The ordinary integro-differential system has been applied to many problems in fluid dynamics, engineering, chemical reactions, and so on.

In the recent years, there are some methods to solve the Fredholm type integro-differential equations [2–8]. At this point, a new method is presented to solve the integro-differential equations. The method is established in reproducing kernel space; the problem of solving the integro-differential problem with a perturbation can be converted into the simple problem of solving the equation. The representation of all the solutions for Fredholm type integro-differential equation is given if it has solutions. The stability is important and references are there in [9, 10]. There are many discussions about the solutions in reproducing kernel space in [11–15]. In this paper, we discuss the Fredholm type integro-differential equation. In the last section, CAS wavelet approximating methods [5], differential transformation methods [6], HPM [1], and reproducing kernel space method are compared, then we can get some effective data. The numerical experiments show that this kind of method is stable in the reproducing kernel space.

2. Two Reproducing Kernel Spaces

2.1. The Reproducing Kernel Space 𝑊²₂[0,1]

The reproducing kernel space 𝑊²₂[0,1] is defined as follows:𝑊²₂[0,1]=𝑢(𝑡)∣𝑢^′(𝑡)isabsolutelycontinuousfunctionin[0,1],𝑢^′′(𝑡)∈ℒ²[0,1]. (2.1)

The inner product and norm in 𝑊²₂[0,1] are defined respectively by⟨𝑢(𝑡),𝑣(𝑡)⟩=¹_𝑖=0𝑢^(𝑖)(0)𝑣^(𝑖)(0)+¹₀𝑢^′′(𝑡)𝑣^′′(𝑡)𝑑𝑡,𝑢,𝑣∈𝑊²₂[],‖𝑢‖0,1_𝑊22[0,1]=⟨𝑢(𝑡),𝑢(𝑡)⟩^1/2,𝑢∈𝑊²₂[0,1]. (2.2)

Then 𝑊²₂[0,1] is a complete reproducing kernel space. That is, there exists a function 𝑅_𝑡(𝑠), for each fixed 𝑡∈[0,1],  𝑅_𝑡(𝑠)∈𝑊²₂[0,1], and for any 𝑢(𝑠)∈𝑊²₂[0,1] and 𝑠∈[0,1], satisfying𝑢(𝑠),𝑅_𝑡(𝑠)_𝑊22=𝑢(𝑡). (2.3) By using Mathematica, 𝑅_𝑡(𝑠) is given by𝑅_𝑡(𝑠)=−16𝑠𝑠²(−1−3𝑡2+𝑠),𝑡≤𝑠,6𝑡³+12𝑡(2+𝑡)𝑠,𝑡>𝑠. (2.4)

2.2. The Reproducing Kernel Space 𝑊¹₂[0,1]

The construction of reproducing kernel space 𝑊¹₂[0,1] can be found in [14] and its reproducing kernel function is𝑄_𝑠(𝑡)=1+𝑡,𝑡≤𝑠,1+𝑠,𝑡>𝑠. (2.5)

3. Analysis of the Solution of (1.1)

Let ℒ∶𝑊²₂[0,1]→𝑊¹₂[0,1], such thatℒ𝑢(𝑡)=𝑢^′(𝑡)−𝑞(𝑡)𝑢(𝑡)−¹₀ℎ(𝑡,𝑠)𝑢(𝑠)𝑑𝑠, (3.1)

where 𝑢(𝑡)∈𝑊²₂[0,1], it is easy to know that ℒ is a linear bounded operator and (1.1) can be converted into the equivalent formℒ𝑢(𝑡)=𝑓(𝑡),𝑡∈[0,1]. (3.2)

In order to obtain the representation of all the solutions of (1.1), let 𝜑_𝑖(𝑡)=𝑄_𝑡𝑖(𝑡), 𝜓_𝑖(𝑡)=ℒ^∗𝜑_𝑖(𝑡), where {𝑡_𝑖}^∞_𝑖=1 is dense in [0,1].

From the definition of the reproducing kernel, we have𝑢(𝑡),𝜑_𝑖(𝑡)_𝑊12=𝑢𝑡_𝑖,𝜓_𝑖(𝑡)=ℒ^∗𝜑_𝑖(𝑠)(𝑡)=ℒ^∗𝑄_𝑡𝑖(𝑠)(𝑎),𝑅_𝑡(𝑎)_𝑊22=𝑄_𝑡𝑖(𝑠),ℒ𝑅_𝑡(𝑎)(𝑠)_𝑊12=ℒ𝑅_𝑡(𝑎)𝑡_𝑖,𝑖=1,2,…, (3.3)

where ℒ^∗ is a conjugate operator of ℒ. Practise Gram-Schmidt orthonormalization for {𝜓_𝑖(𝑡)}^∞_𝑖=1𝜓_𝑖(𝑡)=^𝑖_𝑘=1𝛽_𝑖𝑘𝜓_𝑘(𝑡), (3.4) where 𝛽_𝑖𝑘 are coefficients of Gram-Schmidt orthonormalization and {𝜓_𝑖(𝑡)}^∞_𝑖=1 is orthonormal system in 𝑊²₂[0,1].

Theorem 3.1. If (3.2) has solutions, the results are proved by the following formula: 𝑢(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓𝑡_𝑘𝜓_𝑖(t). (3.5)

Proof. The results are proved by the following formula: 𝑢(𝑡)=^∞_𝑖=1𝑢(𝑡),𝜓_𝑖(𝑡)_𝑊22𝜓_𝑖(𝑡)=^∞_𝑖=1𝑢(𝑡),^𝑖_𝑘=1𝛽_𝑖𝑘𝜓_𝑘(𝑡)_𝑊22𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑢(𝑡),𝜓_𝑘(𝑡)_𝑊22𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑢(𝑡),ℒ^∗𝜑_𝑘(𝑡)_𝑊22𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘ℒ𝑢(𝑡),𝜑_𝑘(𝑡)_𝑊12𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘ℒ𝑢𝑡_𝑘𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓𝑡_𝑘𝜓_𝑖(𝑡). (3.6)

Now, the approximate solution 𝑢_𝑛(𝑡) can be obtained by the 𝑛-term truncation of (3.5), that is𝑢_𝑛(𝑡)=^𝑛𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓𝑡_𝑘𝜓_𝑖(𝑡). (3.7)

Theorem 3.2. Assume 𝑢(𝑡) is the solution of (3.2) and 𝑟_𝑛(𝑡) is the error between the approximate solution 𝑢_𝑛(𝑡) and the exact solution 𝑢(𝑡). Then ‖𝑟_𝑛(𝑡)‖_𝑊²₂→0.

Proof. In the following we prove the sequence 𝑟_𝑛(𝑡) is monotone decreasing in the sense of ‖⋅‖_𝑊²₂.
From (3.5) and (3.7), we have𝑟_𝑛𝑊22=^∞𝑖_{𝑖=𝑛+1𝑘=1}𝛽_𝑖𝑘𝑓𝑡_𝑘𝜓_𝑖(𝑡)_𝑊22=^∞_𝑖=𝑛+1^𝑖_𝑘=1𝛽_𝑖𝑘𝑓𝑡_𝑘²,𝑟_{𝑛−1𝑊22}=^∞𝑖_{𝑖=𝑛𝑘=1}𝛽_𝑖𝑘𝑓𝑡_𝑘𝜓_𝑖(𝑡)_𝑊22=^∞_𝑖=𝑛^𝑖_𝑘=1𝛽_𝑖𝑘𝑓𝑡_𝑘². (3.8) By (3.8), we know that ‖𝑟_𝑛‖_𝑊²₂≤‖𝑟_𝑛−1‖_𝑊²₂. Then {𝑟_𝑛(𝑡)}^∞_𝑛=1 is monotone decreasing in ‖⋅‖_𝑊²₂. Since the series ∑^∞_𝑖=1⟨𝑢(𝑡),𝜓_𝑖(𝑡)⟩_𝑊²₂𝜓_𝑖(𝑡) is convergent in 𝑊²₂[0,1], we obtain ‖𝑟_𝑛(𝑡)‖_𝑊²₂→0.

4. The Stability of the Approximate Solution

In order to consider the stability of the approximate solution, we add a perturbation 𝜀(𝑡) in the right-hand side, then (3.2) becomesℒ𝑢(𝑡)=𝑓(𝑡)+𝜀(𝑡),𝑡∈[0,1]. (4.1)

Now, we discuss the representation of the solution for (4.1).

4.1. Representation of All the Solutions of (4.1)

In order to study the stability of (4.1), let 𝒜 be a projection operator from 𝑊²₂[0,1] to Ψ,Ψ=𝑢∣𝑢=^∞_𝑖=1𝑐_𝑖𝜓_𝑖,for𝑐_𝑖^∞_𝑖=1∈𝔩², (4.2) where {𝜓_𝑖}^∞_𝑖=1 is given in (3.4), and 𝒜 satisfies 𝒜^∗=𝒜. Moreover𝒜𝜓_𝑖(𝑠)=^∞_𝑘=1𝜓_𝑖(𝑠),𝜓_𝑘(𝑠)𝜓_𝑘(𝑠)=𝜓_𝑖(𝑠). (4.3)

We have𝒜𝑢(𝑡)=^∞_𝑖=1𝑢(𝑡),𝜓_𝑖(𝑡)𝜓_𝑖(𝑡), (4.4)

where 𝑢(𝑡) is a solution of (3.2) in 𝑊²₂[0,1].

Define𝑢_𝒜(𝑡)=𝒜𝑢(𝑡), (4.5) then we have𝑢_𝒜(𝑡)=^∞_𝑖=1𝑢(𝑡),𝜓_𝑖(𝑡)𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓𝑡_𝑘𝜓_𝑖(𝑡). (4.6)

Theorem 4.1. If {𝑡_𝑖}^∞_𝑖=1 is dense in [0,1], then the form (4.6) is the solution of (3.2) in Ψ.

Proof. We have ℒ𝑢_𝒜𝑡_𝑖=ℒ𝑢_𝒜(𝑡),𝜑_𝑖(𝑡)=𝑢_𝒜(𝑡),ℒ^∗𝜑_𝑖(𝑡)=𝒜𝑢(𝑡),𝜓_𝑖(𝑡=𝑢(𝑡)),𝒜^∗𝜓_𝑖(𝑡)=𝑢(𝑡),𝒜𝜓_𝑖(𝑡=𝑢(𝑡)),𝜓_𝑖(𝑡)=𝑢(𝑡),ℒ^∗𝜑_𝑖(𝑡=(𝑡))ℒ𝑢,𝜑_𝑖(𝑡)=𝑓(𝑡),𝜑_𝑖(𝑡)=𝑓𝑡_𝑖, (4.7)since {𝑡_𝑖}^∞_𝑖=1 is dense in [0,1], then ℒ𝑢_𝒜(𝑡)=𝑓(𝑡).
It is easy to know that the solution of (3.2) is unique in Ψ (see [16]).

The following lemma holds.

Lemma 4.2. Ψ^⟂=𝑁(ℒ), where Ψ^⟂={𝑢(𝑡)∣⟨𝑢(𝑡),𝑣(𝑡)⟩=0,𝑓𝑜𝑟𝑎𝑛𝑦𝑣(𝑡)∈Ψ} and 𝑁(ℒ) is a null space of ℒ, that is 𝑁(ℒ)={𝑢∣ℒ𝑢=0}.

Proof. For any 𝑢(𝑡)∈Ψ^⟂, 0=𝑢(𝑡),𝜓_𝑘(𝑡)=𝑢(𝑡),ℒ^∗𝜑_𝑘(𝑡)=ℒ𝑢(𝑡),𝜑_𝑘(𝑡)=ℒ𝑢𝑡_𝑘. (4.8)Since {𝑡_𝑖}^∞_𝑖=1 is dense in [0,1], then ℒ𝑢(𝑡)≡0. We can obtain 𝑢(𝑡)∈𝑁(ℒ). Obviously, 𝑁(ℒ)⊂Ψ^⟂.

Then Ψ^⟂=𝑁(ℒ).

The following theorem is obvious.

Theorem 4.3. Let {𝑡_𝑖}^∞_𝑖=1 be any dense subset of [0,1], if (4.1) has solutions, then its solution 𝑢(𝑡) can be represented as follows: 𝑢(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓𝑡_𝑘𝜓_𝑖(𝑡)+𝜏(𝑡), (4.9)where 𝜏(𝑡)∈𝑁(ℒ).

Assume that {𝜎_𝑖}^∞_𝑖=1 is a basis of 𝑁(ℒ). Orthonormalzing {𝜓₁,𝜓₂,…,𝜎₁,𝜎₂,…}, we obtain𝜎_𝑖=^∞_𝑘=1𝛽_𝑖𝑘𝜓_𝑘(𝑡)+^𝑖_𝑗=1𝛽_𝑖𝑗𝜎_𝑗,𝑖=1,2,…. (4.10)

Hence, {𝜓₁,𝜓₂,…,𝜎₁,𝜎₂,…} is a normal orthogonal basis in 𝑊²₂[0,1].

According to Theorem 4.3, we can obtain the solution of (4.1). Hence, through the Gram-Schmidt process, we have {𝜓_𝑖}^∞_𝑖=1∪{𝜎_𝑖}^∞_𝑖=1 are the complete orthonormal system in 𝑊²₂[0,1], and {𝜎_𝑖}^∞_𝑖=1 is a complete orthonormal system in 𝑁(ℒ).

4.2. The Stability of the Solution for (3.2) in Reproducing Kernel Space

Let the space Ψ be complete. ℒ_Ψ be a restricted operator of ℒ in Ψ, we have the converse operator ℒ_Ψ⁻¹∶𝑊²₂[0,1]→Ψ exists and is bounded.

Lemma 4.4. If 𝑢_𝒜(𝑡) is given by (4.6), then 𝑢_𝒜(𝑡) is the minimal norm solution.

Proof. Let 𝑢(𝑡) be a solution of (3.2). We have 𝑢(𝑡)=𝑢_𝒜(𝑡)+𝑣(𝑡), (4.11)where 𝑢_𝒜(𝑡)∈Ψ and 𝑣(𝑡)∈Ψ^⟂. The following ‖𝑢‖²=𝑢_𝒜+𝑣,𝑢_𝒜+𝑣=𝑢_𝒜²+𝑢_𝒜,𝑣+𝑣,𝑢_𝒜+‖𝑣‖²=𝑢_𝒜²+‖𝑣‖²≥𝑢_𝒜² (4.12)holds.
It is pointed that 𝑢_𝒜 is the minimal norm solution of (3.2).

Theorem 4.5. If the (4.1) has solutions and let 𝑢_𝒜(𝑡) be the minimal norm solution, then ‖𝑢_𝒜(𝑡)−𝑢_𝒜,𝑛(𝑡)‖⟶0(𝑛⟶∞), (4.13) where 𝑢_𝒜,𝑛(𝑡) is a truncation of 𝑢_𝒜(𝑡). Hence, 𝑢_𝒜(𝑡) is stable in 𝑊²₂[0,1].

Proof. Let ℒ_Ψ𝑢_𝒜,𝑛(𝑡)=𝑓_𝑛(𝑡) and 𝑓(𝑡)=𝑓_𝑛(𝑡)+𝜀_𝑛(𝑡), where 𝜀_𝑛(𝑡) is a perturbation and 𝜀_𝑛(𝑡)→0(𝑛→∞) in ‖⋅‖_𝑊²₂.
On the one hand, since ℒ_Ψ⁻¹𝜀_𝑛(𝑡)∈Ψ, ℒ^∗_Ψ𝜑_𝑘=𝜓_𝑘, it follows thatℒ_Ψ⁻¹𝜀_𝑛(𝑡)=^∞_𝑖=1ℒ_Ψ⁻¹𝜀_𝑛(𝑡),𝜓_𝑖(𝑡)𝜓_𝑖(𝑡)=^∞_𝑖=1ℒ_Ψ⁻¹𝜀_𝑛(𝑡),^𝑖_𝑘=1𝛽_𝑖𝑘𝜓_𝑘(𝑡)𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘ℒ_Ψ⁻¹𝜀_𝑛(𝑡),𝜓_𝑘(𝑡)𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝜀_𝑛(𝑡),ℒ_Ψ^−1∗𝜓_𝑘(𝑡)𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝜀_𝑛(𝑡),ℒ_Ψ^−1∗ℒ^∗_Ψ𝜑_𝑘(𝑡)𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝜀_𝑛(𝑡),𝜑_𝑘(𝑡)𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝜀_𝑛𝑡_𝑘𝜓_𝑖(𝑡). (4.14)
On the other hand, 𝑓,𝑓_𝑛∈𝑊¹₂[0,1], from the form (4.6), we have𝑢_𝒜(𝑡)−𝑢_𝒜,𝑛(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓(𝑡)−𝑓_𝑛(𝑡)𝜓_𝑖(𝑡)=^∞𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝜀_𝑛𝑡_𝑘𝜓_𝑖(𝑡), (4.15)where 𝑢_𝒜,𝑛(𝑡)=^𝑛𝑖_𝑖=1𝑘=1𝛽_𝑖𝑘𝑓_𝑛𝑡_𝑘𝜓_𝑖(𝑡), (4.16)and then 𝑢_𝒜(𝑡)−𝑢_𝒜,𝑛(𝑡)=ℒ_Ψ⁻¹𝜀_𝑛(𝑡). (4.17)From the continuity of ℒ_Ψ⁻¹ and 𝜀_𝑛(𝑡)→0(𝑛→∞) in ‖⋅‖_𝑊²₂, we have lim_𝑛→∞𝑢_𝒜(𝑡)−𝑢_𝒜𝑛(𝑡)_𝑊22≤ℒ_Ψ⁻¹_𝑊22lim_𝑛→∞𝜀_𝑛(𝑡)_𝑊22=0. (4.18)

5. Numerical Experiments

To illustrate the effectiveness of the above method, we give an example as follows:

Example 5.1. Consider the following first-order Fredholm type integro-differential equation (see [1]): 𝑢^′(𝑡)=(𝑡+1)𝑒^𝑡−𝑡+¹₀(𝑠)[],𝑢(0)𝑡𝑢𝑑𝑠,𝑡∈0,1=0. (5.1)The analytic solution of this equation is 𝑢(𝑡)=𝑡𝑒^𝑡.
From Tables 1, 2, and 3, we can see that the absolute error, relative error and the root-mean-square are small.

Table 1 

Comparison of absolute errors of approximate solutions, for example, obtained by reproducing kernel space methods, CAS wavelet approximating [5], differential transformation [6] methods and HPM [1].

Table 2 

Relative error of approximate solutions with 𝜀=0.0001.

Table 3 

The root-mean-square, for example, with 𝜀=0.0001.

6. Conclusions

From the previous numerical results, we can see that the error is quite small and the numerical solution is stable when the right-hand side with a small perturbation. It illustrates that the method is given in the paper is valid.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (61071181), the Excellent Middle-Young Scientific Research Award Foundation of Heilongjiang Province (QC2010036), academic foundation for youth of Harbin Normal University (10KXQ-05), and science and technology development project of Harbin Normal University (08XYG-13).